Live Exercise 1: Hotelling with quadratic transportation costs

MSc-level Industrial Organisation course at the University of St Andrews

Solutions

Group exercise (≈20 minutes)

• Work in groups of 2–3.
• Show your steps (the algebra is the point).
• Assume full market coverage throughout (everyone buys from one of the two firms).

Problem: Quadratic transport costs and cost asymmetry

Consider a Hotelling line of length 1 with consumers uniformly distributed on \([0,1]\). Each consumer buys one unit from the firm that gives higher utility.

There are two firms with fixed locations:

• Firm 1 at \(x_1=a\)
• Firm 2 at \(x_2=1-a\)

where \(0 \le a < \frac{1}{2}\). Define the distance between firms \(d \equiv x_2-x_1 = 1-2a\).

Consumers have quasi-linear utility

\[ U_i(x) = \bar u - p_i - t(x-x_i)^2, \]

where \(t>0\) and \(\bar u\) is large enough that the outside option never binds. Firm \(i\) has constant marginal cost \(c_i\) and chooses price \(p_i\) simultaneously.

(a) Marginal consumer and demands

Let \(x^*\) be the consumer indifferent between the two firms (assume \(x^* \in (a,1-a)\)).

1. Write the indifference condition defining \(x^*\).
2. Solve for \(x^*\) as a function of \((p_1,p_2,t,a)\).
3. Use your result to write demands \(q_1(p_1,p_2)\) and \(q_2(p_1,p_2)\).

Hint: For \(x \in (a,1-a)\), squared distances are always positive; no absolute values are needed.

(b) Best responses in prices

Firm \(i\)’s profit is

\[ \pi_i(p_i,p_j) = (p_i-c_i)\,q_i(p_i,p_j). \]

1. Take the first-order condition for Firm 1 and simplify it into a linear best response \(BR_1(p_2)\).
2. Do the same for Firm 2.

(c) Price equilibrium (Nash in prices)

Solve the system of best responses to get equilibrium prices \((p_1^*,p_2^*)\). Then compute equilibrium market shares \((q_1^*,q_2^*)\).

1. Show that the price gap satisfies \(p_2^*-p_1^*=\frac{c_2-c_1}{3}\).
2. How does distance \(d\) affect the average markup? (Be explicit.)

(d) When does one firm capture the whole market? (optional challenge)

Your answer in (a)–(c) assumed an interior split (\(x^* \in (a, 1-a)\)).

1. Derive the condition on the cost difference \(\Delta c \equiv c_2-c_1\) (relative to \(t\) and \(d\)) under which the interior split is valid.
2. Give a short economic interpretation: what happens if \(|\Delta c|\) is “too large”?

Suggested solution (sketch)

(a)

Indifference:

\[ \bar u - p_1 - t(x-a)^2 = \bar u - p_2 - t(1-a-x)^2. \]

Cancel \(\bar u\) and rearrange:

\[ p_2-p_1 = t\big[(x-a)^2-(1-a-x)^2\big]. \]

Expanding (or using \((x-a)^2-(1-a-x)^2=(1-2a)(2x-1)=d(2x-1)\)) gives

\[ x^*=\frac{1}{2}+\frac{p_2-p_1}{2td}. \]

For an interior split, demands are

\[ q_1 = x^*, \qquad q_2 = 1-x^*. \]

(b)

Using \(q_1=\frac{1}{2}+\frac{p_2-p_1}{2td}\), we have \(\frac{\partial q_1}{\partial p_1}=-\frac{1}{2td}\). FOC for Firm 1:

\[ 0=\frac{\partial \pi_1}{\partial p_1} = q_1 + (p_1-c_1)\frac{\partial q_1}{\partial p_1} = q_1 - \frac{p_1-c_1}{2td}. \]

So \(p_1-c_1=2td\,q_1\), i.e.

\[ p_1-c_1 = 2td\left(\frac{1}{2}+\frac{p_2-p_1}{2td}\right) = td + p_2 - p_1 \quad \Rightarrow \quad BR_1(p_2)=\frac{p_2+c_1+td}{2}. \]

Similarly,

\[ BR_2(p_1)=\frac{p_1+c_2+td}{2}. \]

(c)

Solve \(2p_1=p_2+c_1+td\) and \(2p_2=p_1+c_2+td\) to get

\[ p_1^* = td + \frac{2c_1+c_2}{3}, \qquad p_2^* = td + \frac{c_1+2c_2}{3}. \]

Price gap: \(p_2^*-p_1^*=\frac{c_2-c_1}{3}\).

Market shares:

\[ q_1^*=\frac{1}{2}+\frac{c_2-c_1}{6td}, \qquad q_2^*=\frac{1}{2}-\frac{c_2-c_1}{6td}. \]

Distance \(d\) raises the common “differentiation” term \(td\) in prices, hence increases average markups and profits.

(d)

Interior split requires \(q_1^*,q_2^* \in (0,1)\), i.e.

\[ \left|\frac{c_2-c_1}{6td}\right| < \frac{1}{2} \quad \Leftrightarrow \quad |c_2-c_1| < 3td. \]

If \(|c_2-c_1|\) is too large relative to \(td\), the low-cost firm can profitably set a price that leaves the rival with (essentially) zero demand.